It depends. If a substance is exposed to a source of neutrons then some nuclei may experience neutron capture and hence produce unstable radioactive isotopes.

However, it is more likely that in these science fiction movies that the person has just experienced some radioactive contamination where traces of the radioactive substance have somehow transferred to their skin, clothing or hair, or worse they have somehow injested some radioactive substance. And these traces are picked up by the geiger counter.

This can have a serious effect on their health, and at large doses human cells will suffer considerable radiation damage but won't make their cells radioactive in themselves.

However, it is more likely that in these science fiction movies that the person has just experienced some radioactive contamination where traces of the radioactive substance have somehow transferred to their skin, clothing or hair, or worse they have somehow injested some radioactive substance. And these traces are picked up by the geiger counter.

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Agreed, from personal experience. The geiger counter doesnt tell you you have radiation poisoning per se. It just tells you if you have radioactive material on/in you. say you breathed in contaminated dust, or that same dust/radioactive mineral specks are on your outer body/clothing.
Holding one of my uranium rock samples isnt going to make/leave me radioactive when I finally place the sample away from me. I am just going to be affected (healthwise) by the particles radiated from the sample whilst I hold it.

Last year (2010) I had the interesting experience of a full body bone scan in hospital. It shows up rheumatoid and osteo arthritis in bone joints. they pumped a large syringe of Technetium99m into me and then did scans. for 2 days I was more radioactive than some of my rock samples. My geiger counter went crazy when brought near me. I even videoed it hahaha, by the 4th day it was almost gone from my system.

Staff: Mentor

However, it is more likely that in these science fiction movies that the person has just experienced some radioactive contamination where traces of the radioactive substance have somehow transferred to their skin, clothing or hair, or worse they have somehow injested some radioactive substance. And these traces are picked up by the geiger counter.

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An even more likely explanation: Its just a movie. A badly done science fiction movie.

Staff: Mentor

For those that have seen them use some sort of meter on people or objects in the news, they are simply searching for radioactive particles released from Japan. It is these particles which make things radioactive.